Quadratic Equation & Linear System Solver (with Steps)
Solve a quadratic equation ax²+bx+c=0 starting from the discriminant, and see whether the roots are real, repeated, or complex. Solve 2- or 3-variable linear systems with Cramer's rule, with no-solution and infinite-solution cases detected. Every step is shown, not just the answer, so it works for homework checks and studying. Everything runs in your browser and nothing you type is sent to a server. For ratios and proportions, try the Ratio & Percentage Calculator.
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1. Quadratic equation ax² + bx + c = 0
Enter the coefficients a, b and c. Blank fields count as 0. If a is 0, the equation is solved as a linear one.
2. Linear system (Cramer's rule)
Choose the number of unknowns, then enter the coefficients and the right-hand constant of each equation. Blank fields count as 0.
Notes
- Calculations use floating-point numbers and results are rounded to 10 significant digits. For linear systems with integer coefficients, non-integer answers are also shown as reduced fractions.
- Quadratic roots are shown as decimals. If you need an exact radical form such as 1 ± √2, read it from the quadratic-formula step.
- When the determinant is 0, the ranks of the coefficient matrix and the augmented matrix are compared to tell "no solution" from "infinitely many solutions".
- This tool is for learning and checking work. Verify the results yourself before using them in an exam or at work.
Glossary
Discriminant
For ax²+bx+c=0 it is D = b² − 4ac. If D > 0 there are two distinct real roots, if D = 0 one repeated root, and if D < 0 two complex conjugate roots.
Repeated root
A quadratic whose two roots coincide. Its parabola just touches the x-axis, and the root is x = −b / 2a.
Complex root
A root containing the imaginary unit i (i² = −1). For real coefficients and D < 0, the roots are the conjugate pair (−b ± √|D| i) / 2a.
Cramer's rule
When the determinant D of the coefficient matrix is not 0, each unknown equals the determinant with its column replaced by the constants, divided by D. It suits 2- and 3-variable systems.
FAQ
How does the discriminant decide the type of roots?
With D = b² − 4ac, a positive D gives two distinct real roots, zero gives one repeated root, and a negative D gives two complex conjugate roots. The steps start from the discriminant.
What happens if I set a to 0?
With a = 0 the equation is no longer quadratic, so it is solved as bx + c = 0. If b is also 0, every x is a solution when c = 0, and there is no solution when c ≠ 0.
How can I tell "no solution" from "infinitely many solutions" in a linear system?
When the determinant is 0, the tool compares the rank of the coefficient matrix with the rank of the augmented matrix. Different ranks mean no solution; equal ranks mean infinitely many. Both ranks appear in the steps.
How are fractional answers to a linear system shown?
If every coefficient is an integer, non-integer answers are shown both as a reduced fraction (such as 2/3) and as a decimal. With decimal coefficients, only a decimal rounded to 10 significant digits is shown.
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